Theorem madebdayim 28279

Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
madebdayim	⊢ (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6919	. . 3 ⊢ (𝑋 ∈ (M‘𝐴) → 𝐴 ∈ dom M)
2		madef 28227	. . . 4 ⊢ M:On⟶𝒫 No
3	2	fdmi 6721	. . 3 ⊢ dom M = On
4	1, 3	eleqtrdi 2870	. 2 ⊢ (𝑋 ∈ (M‘𝐴) → 𝐴 ∈ On)
5		fveq2 6885	. . . . . 6 ⊢ (𝑎 = 𝑏 → (M‘𝑎) = (M‘𝑏))
6		sseq2 3934	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝑏))
7	5, 6	raleqbidv 3334	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ (M‘𝑏)(bday‘𝑥) ⊆ 𝑏))
8		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (bday‘𝑥) = (bday‘𝑦))
9	8	sseq1d 3939	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((bday‘𝑥) ⊆ 𝑏 ↔ (bday‘𝑦) ⊆ 𝑏))
10	9	cbvralvw 3240	. . . . 5 ⊢ (∀𝑥 ∈ (M‘𝑏)(bday‘𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏)
11	7, 10	bitrdi 290	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏))
12		fveq2 6885	. . . . 5 ⊢ (𝑎 = 𝐴 → (M‘𝑎) = (M‘𝐴))
13		sseq2 3934	. . . . 5 ⊢ (𝑎 = 𝐴 → ((bday‘𝑥) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝐴))
14	12, 13	raleqbidv 3334	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴))
15		elmade2 28249	. . . . . . . 8 ⊢ (𝑎 ∈ On → (𝑥 ∈ (M‘𝑎) ↔ ∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑥 ∈ (M‘𝑎) ↔ ∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17		elpwi 4563	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝒫 (O‘𝑎) → 𝑙 ⊆ (O‘𝑎))
18		elpwi 4563	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝒫 (O‘𝑎) → 𝑟 ⊆ (O‘𝑎))
19	17, 18	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → (𝑙 ⊆ (O‘𝑎) ∧ 𝑟 ⊆ (O‘𝑎)))
20		unss 4188	. . . . . . . . . 10 ⊢ ((𝑙 ⊆ (O‘𝑎) ∧ 𝑟 ⊆ (O‘𝑎)) ↔ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎))
21	19, 20	sylib 221	. . . . . . . . 9 ⊢ ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → (𝑙 ∪ 𝑟) ⊆ (O‘𝑎))
22		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23		simplll 787	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24		dfss3 3897	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ (O‘𝑎))
25		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (bday‘𝑦) = (bday‘𝑧))
26	25	sseq1d 3939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((bday‘𝑦) ⊆ 𝑏 ↔ (bday‘𝑧) ⊆ 𝑏))
27	26	rspccv 3573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏))
28	27	ralimi 3099	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → ∀𝑏 ∈ 𝑎 (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏))
29		rexim 3103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 (𝑧 ∈ (M‘𝑏) → (bday‘𝑧) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
30	28, 29	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
31	30	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏) → ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
32		elold 28250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (𝑧 ∈ (O‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏)))
33	32	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑧 ∈ (O‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ (M‘𝑏)))
34		bdayon 28143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday‘𝑧) ∈ On
35		onelssex 6412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((bday‘𝑧) ∈ On ∧ 𝑎 ∈ On) → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
36	34, 35	mpan 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
37	36	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((bday‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 (bday‘𝑧) ⊆ 𝑏))
38	31, 33, 37	3imtr4d 297	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑧 ∈ (O‘𝑎) → (bday‘𝑧) ∈ 𝑎))
39	38	ralimdv 3176	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ (O‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
40	24, 39	biimtrid 245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
41	40	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎)
42	41	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎)
43		bdayfun 28138	. . . . . . . . . . . . . . . . 17 ⊢ Fun bday
44		oldssno 28232	. . . . . . . . . . . . . . . . . . 19 ⊢ (O‘𝑎) ⊆ No
45		sstr 3916	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) ∧ (O‘𝑎) ⊆ No) → (𝑙 ∪ 𝑟) ⊆ No)
46	44, 45	mpan2 704	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → (𝑙 ∪ 𝑟) ⊆ No)
47		bdaydm 28140	. . . . . . . . . . . . . . . . . 18 ⊢ dom bday = No
48	46, 47	sseqtrrdi 3949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → (𝑙 ∪ 𝑟) ⊆ dom bday)
49		funimass4 6949	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun bday ∧ (𝑙 ∪ 𝑟) ⊆ dom bday) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
50	43, 48, 49	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
51	50	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
52	51	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)(bday‘𝑧) ∈ 𝑎))
53	42, 52	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → (bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎)
54		cutbdaybnd 28186	. . . . . . . . . . . . 13 ⊢ ((𝑙 <<s 𝑟 ∧ 𝑎 ∈ On ∧ (bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎) → (bday‘(𝑙 |s 𝑟)) ⊆ 𝑎)
55	22, 23, 53, 54	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → (bday‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56		fveq2 6885	. . . . . . . . . . . . 13 ⊢ ((𝑙 |s 𝑟) = 𝑥 → (bday‘(𝑙 |s 𝑟)) = (bday‘𝑥))
57	56	sseq1d 3939	. . . . . . . . . . . 12 ⊢ ((𝑙 |s 𝑟) = 𝑥 → ((bday‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ (bday‘𝑥) ⊆ 𝑎))
58	55, 57	syl5ibcom 248	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → (bday‘𝑥) ⊆ 𝑎))
59	58	expimpd 459	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ (O‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎))
60	59	ex 418	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ (O‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎)))
61	21, 60	syl5 35	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 (O‘𝑎) ∧ 𝑟 ∈ 𝒫 (O‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎)))
62	61	rexlimdvv 3218	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 (O‘𝑎)∃𝑟 ∈ 𝒫 (O‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → (bday‘𝑥) ⊆ 𝑎))
63	16, 62	sylbid 243	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → (𝑥 ∈ (M‘𝑎) → (bday‘𝑥) ⊆ 𝑎))
64	63	ralrimiv 3153	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏) → ∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎)
65	64	ex 418	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ (M‘𝑏)(bday‘𝑦) ⊆ 𝑏 → ∀𝑥 ∈ (M‘𝑎)(bday‘𝑥) ⊆ 𝑎))
66	11, 14, 65	tfis3 7869	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴)
67		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝑋 → (bday‘𝑥) = (bday‘𝑋))
68	67	sseq1d 3939	. . . 4 ⊢ (𝑥 = 𝑋 → ((bday‘𝑥) ⊆ 𝐴 ↔ (bday‘𝑋) ⊆ 𝐴))
69	68	rspccv 3573	. . 3 ⊢ (∀𝑥 ∈ (M‘𝐴)(bday‘𝑥) ⊆ 𝐴 → (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴))
70	66, 69	syl 18	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴))
71	4, 70	mpcom 39	1 ⊢ (𝑋 ∈ (M‘𝐴) → (bday‘𝑋) ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149  𝒫 cpw 4556   class class class wbr 5102  dom cdm 5650   “ cima 5653  Oncon0 6362  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150  Mcmade 28213  Ocold 28214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219

This theorem is used by:  oldbdayim  28280  madebday  28291  onsbnd  28672  onsbnd2  28673